Stably exotic 4-manifolds

A pair of closed, smooth $4$-manifolds $M$ and $M'$ are stably exotic if they are stably homeomorphic but not stably diffeomorphic, where stabilisation refers to connected sum with copies of $S^2 \times S^2$. Orientable stable exotica do not exist by a result of Gompf, but Kreck showed that nonorientable examples are plentiful. We investigate which values of the fundamental group $\pi$ and the first and second Stiefel-Whitney classes $w_1$ and $w_2$ admit stably exotic pairs, providing a complete description if $H_5(\pi;\mathbb{Z})=0$. In particular we produce new stable exotica, and new settings in which they do not arise.